Abstract. Advanced type system features, such as GADTs, type classes, and type families have have proven to be invaluable language extensions for ensuring data invariants and program correctness among others. Unfortunately, they pose a tough problem for type inference, because they introduce local type assumptions.

In this article we present a novel constraint-based type inference approach for local type assumptions. Our system, called OutsideIn(X), is parameterised over the particular underlying constraint domain X, in the same way as HM(X). This stratification allows us to use a common metatheory and inference algorithm.

Going beyond the general framework, we also give a particular constraint solver for X = type classes + GADTs + type families, a non-trivial challenge in its own right.

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Batterseapower 14:24, 14 May 2010 (UTC) You say that local definitions such as "combine" and "swap" could harmlessly be defined at top level. However, there is a cost to doing so in that it pollutes the top-level namespace. Adding a facility for making definitions that are at the top level for the purposes of type checking but not for name resolution would fix this, at the cost of some ugliness.

Figure 2: VarCon uses \nu in the conclusion and x in the premises, it seems they should both be \nu considering the textual description below, but then the definition of "Type environments" in Figure 1 also needs to have \nu rather than x.